where \( b\)  describes the magnetic field.
For this ideal MHD system, the cross-helicity
\(\int _{\Omega } v(x,t)\cdot h(x,t)dx\)  is conserved for the smooth solutions. To the knowledge of the authors, Yu [1]} obtained cross-helicity conservation criterion of weak solutions for ideal MHD equations based on the condition that
\(v\in L^{3}(0,T;B^{\alpha _{1}}_{3,\infty })\)  and \(b\in L^{3}(0,T;B^{\alpha _{2}}_{3, c(\mathbb {N})})\)  with \(\alpha _{1}+2\alpha _{2}\ge 1\)  and \(\alpha _{2}\ge 3\)  (see also [2]}, [3]}, [4]}).
Inspired by Yu's work [1]}, we consider the helicity conservation of weak solutions to the incompressible Euler equations (REF ) in spaces \(B^{\theta }_{p,c(\mathbb {N})}\)  and \(B^{\theta }_{p,\infty }\) . It should be pointed out that this coincides
with the appearance of helicity conservation of weak solutions to the compressible Euler equations (REF ) motivated by the cross-helicity conservation in the MHD system (see [6]}, [7]}). The proof of (1) in Theorem REF  and (1)-(2) in Theorem REF  is very closed to that in [8]}, [1]}. The key point is our new observation that
functions in the Onsager type spaces \(\dot{B}^{1/3}_{p,c(\mathbb {N})}\)  can satisfy the corresponding Constantin-E-Titi type commutator estimates in physical spaces (see Lemma REF ).
It seems that the argument presented here is quite general and can be applied in other fluid equations.
Particularly, we shall show that our approach here can be applied in the 2D surface quasi-geostrophic (SQG) equation below,
\(\left\lbrace \begin{aligned}&\theta _{t} + v\cdot \nabla \theta =0, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{in } ~(0,T)\times \Omega ,\\&v=\nabla ^\perp \Psi =(-\frac{\partial \Psi }{\partial _{x_2}},\frac{\partial \Psi }{\partial _{x_1}}),~\theta =-(-\Delta )^{\frac{1}{2}} \Psi , ~~~~~\text{in } ~[0,T)\times \Omega ,\\&\theta |_{t=0}=\theta _0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{in } ~\Omega ,\end{aligned}\right.\) 
